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Home > Theory > Stress Post- Processing > Stress in Flexible Beams

Stress in flexible beams

The panel for the flexible beam properties is as follows:

  • This panel is typically used for flexible risers or umbilicals for which the internal cross section is complex and, most of the time, stresses cant be derived from the pipe main dimensions (OD and WT).

  • This panel can also be useful to simulate a bundle as an equivalent pipe with global mechanical properties (panel top left side) and post-process the stresses in one pipe which composed this bundle (case stress post-processing only).

The leading assumption is that the axial strain and the curvature of the bundle may be applied to each individual component. Therefore, the stress post-processing is chained as such:

  • Derive the strain and curvature of the pipe from the nodes positions stored in the database files (DTBS, DTB or DTBF);

  • Calculate the mechanical properties of the pipe using its outer diameter, its thickness and the young modulus defined in the panel;

  • Derive the tensions and moments in the pipe;

  • Take into account the potential thickness corrosion to re-evaluate the pipe steel aera and inertia;

  • Derive the stresses as described in the axial and bending stress topic.

Note (1):

It is important to highlight that the real mechanical stifness of the bundle are only used to solve the global equations of motions to get the node positions at equilibrium in static or dynamic.

At the post-processing stage, these efforts are calculated for the individual pipe alone based on the strain and curvature.

As a consequence, the tension post-processed at one line end may be different from the reaction force at this very end since the reaction force is directly provided by the solver during the calculations.

Note (2):

When the option stress post-processing only is selected, the end cap effect due to the inner and outer pressures is disregarded to be consistent with the fact that the pressure effect on the beam strain has not been taken into account during the calculations.

In that case, the Poisson coefficient value is not used.

Indeed, let \(\sigma_t\) and \(\sigma_\gamma\) be the tangential and the radial stresses due to pressure. The true axial stress \(\sigma_{real}\) is deduced from the effective stress \(\sigma_{eff}\): $$ \sigma_{real}=\sigma_{eff}+\sigma_p $$

with:

\[ \sigma_p=\frac{P_iS_i-P_eS_e}{S_e-S_i} \]

On another hand, due to the Poisons coefficient, the fluid pressures have an effect on the axial strain. The well-known expressions of Lam actually write: $$ \epsilon_{ij}=\frac{1+\nu}{E}\sigma_{ij}-\frac{\nu}{E}\delta_{ij}(\sigma_{11}+\sigma_{22}+\sigma_{33}) $$

With this formula, the axial strain may be expressed as a function of the the true axial stress \(\sigma_{real}\), the tangential stress and the radial stress \(\sigma_\gamma\) : $$ \epsilon_{axial}=\frac{1}{E}\sigma_{real}-\frac{\nu}{E}(\sigma_{t}+\sigma_\gamma)=\frac{1}{E}\sigma_{real}-\frac{2\nu}{E}\sigma_{\gamma} $$

Note (3):

For beams with bidirectional bending stiffness, the post-processing is slightly different form what is described above, but the results is the same. Derive the strain and curvature of the pipe from the nodes positions stored in the databse files (DTBS, DTB or DTBF); Derive the tensions and moments in the pipe using the bidirectional bending striffnesses; Calculate the mechanical properties of the equivalent circular beam with its outer diameter, its thickness and the young modulus defined in the panel; Correct the bending moment vector with the ratios \(\frac{EI_{equiv}}{EI_1}\) and \(\frac{EI_{equiv}}{EI_2}\); Take into account the potential thickness corrosion to re-evaluate the pipe steel aera and inertia; Derive the stresses as described in the axial and bending stress topic.